Characterization of some finite simple groups by the set of orders of vanishing elements and order

S. Askary

doi:10.37863/umzh.v73i11.1069

S. Askary Imam-Ali Univ., Tehran, Iran

DOI: https://doi.org/10.37863/umzh.v73i11.1069

Keywords: Vanishing element, finite simple groups, Vanishing prime graph

Abstract

UDC 512.5

Характеризацiя деяких скiнченних простих груп множиною порядкiв зникаючих елементiв та порядку

Let $G$ be a finite group. We say that an element $g$ of $G$ is a vanishing element if there exists an irreducible complex character $X$ of $G$ such that $X(g) = 0$. Ghasemabadi, Iranmanesh, Mavadatpour (2015), present the following conjecture: Let $G$ be a finite group and $M$ a finite nonabelian simple group such that $Vo(G)=Vo(M)$ and $|G|=|M|$. Then $G \cong M $. We answer in affirmative this conjecture for $M = ^2 D_{r+1}(2)$, where $r = 2^n - 1 \geq 3$ and either $2^r+1$ or $2^{r+1}+1$ is a prime number and $M = ^2 D_{r}(3)$, where $r = 2^n + 1 \geq 5$ and either $(3^{r-1}+1)/2$ or $(3^{r}+1)/4$ is prime.

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